and its generating function \(\mathcal {F}_{\tau }(u,v):= (-2)^t \sum _{k\ge 2}\widetilde{{G}_{\mathbb {F}, {k}}}(\tau , \frac{ u v }{ 2\pi i } )( \mathcal {N}(u)^{ {k-1}}+\mathcal {N}(v)^{ {k-1}})\)  behaves as a Jacobi-like form [1]}, [2]} with a modular transformation property
\( \mathcal {F}_{\frac{a\tau +b}{c\tau +d}}(\frac{u}{c\tau +d}, \frac{v}{c\tau +d})=\mathcal {N}(c\tau +d)\, e^{ tr(\frac{cuv}{ c\tau +d })}{\mathcal {F}}_{\tau }(u,v), \forall \left({\begin{matrix}a&b\\c&d\end{matrix}}\right)\in \Gamma . \) 

